Expressing 5⁴ × 6⁸ As A Power With Exponent 4
Hey guys! Let's dive into a cool math problem today. We're going to figure out how to express the expression 5⁴ × 6⁸ as a power with the exponent 4. This might sound a bit tricky at first, but trust me, it's totally doable. We'll break it down step by step, so you can follow along easily. Get ready to flex those mathematical muscles!
Understanding the Basics of Exponents
Before we jump into the problem, let's quickly refresh our understanding of exponents. An exponent tells you how many times a number (the base) is multiplied by itself. For example, in the expression 2³, the base is 2 and the exponent is 3. This means we multiply 2 by itself three times: 2 × 2 × 2 = 8. Got it? Great!
Now, when we're dealing with expressions like 5⁴ × 6⁸, we need to remember a few key rules of exponents. One important rule is that when we multiply numbers with the same base, we can add their exponents. However, in our case, the bases are different (5 and 6), so we can't directly apply this rule. Instead, we need to find a way to rewrite the expression so that we can express it with the desired exponent of 4. This involves a bit of algebraic manipulation, but don't worry, we'll take it one step at a time. Let's first look at each component separately and see how we can massage them into the form we need.
Breaking down the problem into smaller, manageable parts is always a good strategy in math. It helps to make the problem less intimidating and allows us to focus on each aspect individually. So, let’s get to it and start breaking down those exponents!
Deconstructing 5⁴ × 6⁸
Okay, let's tackle our expression: 5⁴ × 6⁸. The goal here is to rewrite this in the form of something raised to the power of 4. We already have 5 raised to the power of 4, which is a great start! But what about 6⁸? We need to figure out how to express 6⁸ in terms of an exponent of 4.
Think of it this way: we want to find a number that, when raised to the power of 4, gives us 6⁸. This involves understanding how exponents multiply when you have a power raised to another power. Remember the rule: (aᵇ)ᶜ = aᵇᶜ? This is exactly what we need here. We want to find a 'b' such that 4 times 'b' equals 8. Simple enough, right? 4 × 2 = 8. So, we need to rewrite 6⁸ as a power of something squared, all raised to the power of 4.
So, 6⁸ can be thought of as (6²)⁴. Now, what is 6²? It's 6 × 6, which equals 36. So we can rewrite 6⁸ as (36)⁴. Now our original expression looks like this: 5⁴ × (36)⁴. See how we're getting closer? We now have two terms, both raised to the power of 4. This is excellent progress!
The next step involves using another rule of exponents: aⁿ × bⁿ = (a × b)ⁿ. This rule allows us to combine two numbers raised to the same power by multiplying their bases and keeping the exponent the same. This is the key to solving our problem and expressing the entire expression as a single power with an exponent of 4. Let’s put this rule into action and see how it simplifies our expression further.
Applying the Power of a Product Rule
Alright, now that we've massaged our expression into 5⁴ × (36)⁴, we're in the perfect position to apply the power of a product rule. This rule, as we mentioned before, states that aⁿ × bⁿ = (a × b)ⁿ. In our case, 'a' is 5, 'b' is 36, and 'n' is 4. So, let’s plug these values into the formula and see what we get.
Applying the rule, we get (5 × 36)⁴. Now we just need to multiply 5 by 36. Grab your calculators, or do it the old-fashioned way – either way, 5 × 36 equals 180. So now we have (180)⁴.
And there we have it! We've successfully expressed 5⁴ × 6⁸ as a single power with the exponent 4. The answer is 180⁴. Wasn't that satisfying? We took a seemingly complex problem and broke it down into manageable steps, using the rules of exponents to guide us. This is the beauty of math – taking something complicated and making it simple through logical steps.
This process highlights the importance of understanding the rules of exponents and how they can be used to manipulate expressions. By recognizing the patterns and applying the correct rules, we can simplify even the most daunting mathematical challenges. So, remember this approach whenever you encounter similar problems – break it down, apply the rules, and solve!
Final Answer and Conclusion
So, to recap, we started with the expression 5⁴ × 6⁸ and our mission was to express it as a power with the exponent 4. We deconstructed 6⁸, recognizing it could be rewritten as (36)⁴. Then, we applied the power of a product rule to combine the bases, resulting in (5 × 36)⁴. Finally, we multiplied 5 by 36 to get 180, giving us our final answer:
5⁴ × 6⁸ = 180⁴
See, math isn’t so scary when you break it down! By understanding the basic rules and applying them step by step, you can tackle even the trickiest problems. Remember, the key is to practice and be patient with yourself. Keep exploring, keep questioning, and keep having fun with math!
I hope this explanation helped you guys understand how to express 5⁴ × 6⁸ as a power with the exponent 4. If you have any more questions or math problems you'd like me to tackle, just let me know. Happy calculating! Keep the math magic flowing, and I'll catch you in the next problem-solving adventure!